{
  "id": "1503.04156",
  "version": "v1",
  "published": "2015-03-13T17:28:01.000Z",
  "updated": "2015-03-13T17:28:01.000Z",
  "title": "Local Limit Theorem in negative curvature",
  "authors": [
    "François Ledrappier",
    "Seonhee Lim"
  ],
  "comment": "52 pages, 5 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\\lim_{t \\to \\infty} t^{3/2}e^{\\lambda_0 t} p(t,x,y)=C(x,y),$$ where $\\lambda_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x, y \\in X$. We also show that the $\\lambda_0$-Martin boundary of $X$ is equal to its topological boundary. We use the uniform Harnack inequality on the boundary $\\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-13T17:28:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40",
      "37A17",
      "37A25",
      "37A30",
      "37A50"
    ],
    "keywords": [
      "local limit theorem",
      "negative curvature",
      "uniform harnack inequality",
      "unit tangent bundle",
      "martin boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150304156L"
    }
  }
}